The Topology of Open Manifolds with Nonnegative Ricci Curvature

Research Foundation Of The City University Of New York (Lehman), Bronx NY

Investigators

Abstract

Abstract for NSF Proposal DMS - 0102279 The Topology of Open Manifolds with Nonnegative Ricci Curvature Christina Sormani Dr. Sormani proposes to study the topology of complete manifolds with nonnegative Ricci curvature and their limit spaces. In particular she plans to investigate various approaches to Milnor's conjecture that the fundamental group of an open manifold with nonnegative Ricci curvature is finitely generated. She also plans to study the higher dimensional homology of these spaces. Techniques which will be employed involve Gromov-Hausdorff limits, the almost rigidity theory of Cheeger-Colding, and Busemann functions. In particular, the properness of Busemann functions on these manifolds will be investigated. It should be noted that there are direct applications of this project to the theory of topological censorship in general relativity. The condition of nonnegative Ricci curvature on space-time is called the null energy condition and it arises in the Einstein equation. Roughly speaking, Dr. Sormani proposes to study the existence and prevalence of holes in a space which has no boundary, extends to infinity and has a condition imposed upon the way in which it can bend. The universe we live in is such a space. Simpler examples are cylinders (i.e. tubes) and paraboloids (i.e. bowls). The cylinder has a hole but the paraboloid does not. The spaces studied in this project are of arbitrary dimension and so the holes come in various dimensions as well. The universe is one such higher dimensional space and its holes, which may or may not exist, are often called wormholes. By furthering our understanding of this geometric problem, it is hoped that we will further our understanding of the universe.